Finite Group Databasebeta

← #14540324

Subgroups — #14540324

Order 2176.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587600C211yesmaximal, characteristic, complemented
2#11763478C211yesmaximal, characteristic, complemented
3#11763447C211yesmaximal, characteristic, complemented
4#11763195C211yesmaximal, characteristic, complemented
5#11763052C211yesmaximal, characteristic, complemented
6#11763045C211yesmaximal, characteristic, complemented
7#11763033C211yesmaximal, characteristic, complemented
8#11763002C211yesmaximal, characteristic, complemented
9#11762970C211yesmaximal, characteristic, complemented
10#11762954C211yesmaximal, characteristic, complemented
11#11762954C211yesmaximal, characteristic, complemented
12#11762941C211yesmaximal, characteristic, complemented
13#11762910C211yesmaximal, characteristic, complemented
14#11762880C211yesmaximal, characteristic, complemented
15C2×C17×#135C211yesmaximal, characteristic, complemented
16C2×#41×C17C2211yescharacteristic
17C2×#41×C17C2211yescharacteristic, complemented
18C2×#63256C2211yescharacteristic
19C2×D4×D17C2211yescharacteristic
20C17×#135—24no—
21C2×Q16×C17C2211yescharacteristic, complemented
22C2×SD16×C17C2211yescharacteristic
23C2×SD16×C17C2211yescharacteristic, complemented
24C2×D8×C17C2211yescharacteristic
25C22×C8×C17C2211yescharacteristic, complemented
26#10587633C2211yescharacteristic
27#10587632C2211yescharacteristic
28Q8×Dic17C2211yescharacteristic, complemented
29C2×#63230C2211yescharacteristic
30C2×#63230C2211yescharacteristic, complemented
31#10587626—21no—
32#10587626C2211yescharacteristic, complemented
33#10587625—21no—
34#10587624—21no—
35#10587624C2211yescharacteristic
36#10587623—21no—
37D4×Dic17C2211yescharacteristic
38C2×#63228C2211yescharacteristic
39C2×#63228C2211yescharacteristic
40#10587616—21no—
41#10587616C2211yescharacteristic
42#10587616C2211yescharacteristic, complemented
43#10587615—21no—
44C4×C22⋊D17—21no—
45C4×C22⋊D17C2211yescharacteristic
46C2×#63227—21no—
47#10587600—24no—
48C2×D136C2211yescharacteristic
49C2×#63218C2211yescharacteristic
50#10587506C2211yescharacteristic, complemented
51#10587506C2211yescharacteristic, complemented
52#10587503C2211yescharacteristic
53#10587503C2211yescharacteristic, complemented
54C22⋊#3622C2211yescharacteristic, complemented
55#10587492C2211yescharacteristic
56#10587492C2211yescharacteristic, complemented
57#10587491C2211yescharacteristic
58#10587489C2211yescharacteristic, complemented
59#10587486C2211yescharacteristic, complemented
60#10587485C2211yescharacteristic, complemented
61C23×D17—21no—
62#41×C17—22no—
63#41×C17—22no—
64#41×C17—41no—
65#41×C17—41no—
66C2×Q8×C17C2311yescharacteristic
67C2×Q8×C17C2311yescharacteristic
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17C2311yescharacteristic
71C2×D4×C17C2311yescharacteristic
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17C2311yescharacteristic
75C2×C22⋊D17—21no—
76C2×C22⋊D17—21no—
77C2×C22⋊D17—21no—
78C2×C22⋊D17—21no—
79C2×C22⋊D17—21no—
80C2×C22⋊D17—21no—
81C2×C22⋊D17D411yescharacteristic
82C22×Dic17—21no—
83#63256—22no—
84#63256—41no—
85D4×D17—22no—
86D4×D17—41no—
87C2×D68—21no—
88C2×D68C2311yescharacteristic
89C2×D68C2311yescharacteristic
90C2×C4×D17—21no—
91C2×C4×D17C2311yescharacteristic
92Q16×C17—22no—
93SD16×C17—22no—
94SD16×C17D414yescomplemented
95D8×C17—22no—
96C2×C8×C17—22no—
97C2×C8×C17C2311yescharacteristic
98C2×C8×C17C2311yescharacteristic
99C22⋊Dic17—21no—
100C22⋊Dic17—21no—
101C22⋊Dic17D411yescharacteristic
102#63230—22no—
103#63230—41no—
104#63228—22no—
105#63228—41no—
106#63227—21no—
107#63227—21no—
108#63227—21no—
109#63227—21no—
110#63227—21no—
111#63227D411yescharacteristic
112C4⋊Dic17—21no—
113C4⋊Dic17C2311yescharacteristic
114C4⋊Dic17C2311yescharacteristic
115#63225—21no—
116#63225D411yescharacteristic
117C4×Dic17—21no—
118C4×Dic17C2311yescharacteristic
119C2×#3622C2311yescharacteristic
120C2×#3622C2311yescharacteristic
121D136—22no—
122#63218—22no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17C2×D411yescharacteristic
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—42no—
129C22×D17—42no—
130C22×D17—81no—
131C22×D17C2×D411yescharacteristic
132Q8×C17—21no—
133Q8×C17—21no—
134Q8×C17—21no—
135Q8×C17C2×D412yes—
136D4×C17—21no—
137D4×C17—21no—
138D4×C17—21no—
139D4×C17—42no—
140D4×C17—42no—
141D4×C17C2×D412yes—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17C2×D411yescharacteristic
149C2×C4×C17C2411yescharacteristic
150C22⋊D17—41no—
151C22⋊D17—42no—
152C22⋊D17—42no—
153C22⋊D17—81no—
154C22⋊D17—81no—
155C22⋊D17—81no—
156C22⋊D17—81no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—81no—
160C2×Dic17C2×D411yescharacteristic
161C2×Dic17C2×D411yescharacteristic
162C2×Dic17C2×D411yescharacteristic
163D68—21no—
164D68—21no—
165D68—21no—
166D68—21no—
167D68—42no—
168C4×D17—21no—
169C4×D17—21no—
170C4×D17—42no—
171C8×C17—21no—
172C8×C17C2×D412yes—
173#3622—21no—
174#3622—21no—
175#3268—171nomaximal, Sylow
176C22×C17—22no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17C22×D411yescharacteristic
186C2×D17—21no—
187C2×D17—21no—
188C2×D17—21no—
189C2×D17—41no—
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—84no—
196C4×C17—22no—
197C4×C17—21no—
198C4×C17—41no—
199C4×C17C22×D411yescharacteristic
200C4×C17H(5,𝔽2)11yescharacteristic
201Dic17—21no—
202Dic17—21no—
203Dic17—21no—
204Dic17—41no—
205Dic17—41no—
206C2×AGL(1,ℤ/8)—171no—
207C2×#135D1711yescharacteristic, complemented
208#550—171no—
209#546—171no—
210#496—171no—
211#468—171no—
212#466—171no—
213#464—171no—
214#459—171no—
215#452—171no—
216#449—171no—
217#449—171no—
218#447—171no—
219#440—171no—
220#435—171no—
221C2×C17—42no—
222C2×C17—41no—
223C2×C17—81no—
224C2×C17D4211yescharacteristic
225C2×C17#57511yescharacteristic
226C2×C17#57611yescharacteristic
227D17—41no—
228D17—81no—
229D17—81no—
230C2×#41—171no—
231C2×#41C2×D1711yescharacteristic
232C2×#41C2×D1711yescharacteristic, complemented
233C22×D4—171no—
234AGL(1,ℤ/8)—344no—
235#135—24no—
236C2×Q16C2×D1711yescharacteristic, complemented
237C2×SD16—171no—
238C2×SD16—171no—
239C2×SD16C2×D1711yescharacteristic
240C2×SD16C2×D1711yescharacteristic, complemented
241C2×D8—171no—
242C2×D8—171no—
243C2×D8—171no—
244C2×D8C2×D1711yescharacteristic
245C2×M16—171no—
246C22×C8C2×D1711yescharacteristic, complemented
247#127—171no—
248#127—341no—
249#124—171no—
250#123—341no—
251C22⋊Q8—171no—
252#121—171no—
253#121—171no—
254#121—171no—
255#121—341no—
256#121—341no—
257#121—341no—
258#120—341no—
259C4×Q8—171no—
260C4×D4—171no—
261C4×D4—171no—
262C4×D4—341no—
263C2×C22⋊C4—341no—
264#107—171no—
265C4⋊C8—171no—
266#103—171no—
267#103—171no—
268#102—171no—
269#102—171no—
270#102—171no—
271#102—171no—
272C22⋊C8—171no—
273C22⋊C8—171no—
274#97—171no—
275C17#326811yescharacteristic, complemented, Sylow
276C24—341no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—342no—
282#41—681no—
283C2×Q8C22×D1711yescharacteristic
284C2×Q8C22×D1711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—171no—
288C2×D4—171no—
289C2×D4—171no—
290C2×D4—341no—
291C2×D4—341no—
292C2×D4—341no—
293C2×D4—341no—
294C2×D4—342no—
295C2×D4—341no—
296C2×D4—341no—
297C2×D4—341no—
298C2×D4—681no—
299C2×D4C22×D1711yescharacteristic
300C2×D4C22×D1711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—171no—
304C22×C4—341no—
305C22×C4—341no—
306C22×C4C22×D1711yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—342no—
310SD16—681no—
311SD16C22⋊D1714yescomplemented
312D8—22no—
313D8—342no—
314D8—342no—
315D8—681no—
316M16—342no—
317C2×C8—22no—
318C2×C8—171no—
319C2×C8—171no—
320C2×C8C22×D1711yescharacteristic
321C2×C8C22×D1711yescharacteristic
322C4⋊C4—171no—
323C4⋊C4—171no—
324C4⋊C4—171no—
325C4⋊C4—341no—
326C4⋊C4—341no—
327C22⋊C4—171no—
328C22⋊C4—171no—
329C22⋊C4—341no—
330C22⋊C4—341no—
331C22⋊C4—341no—
332C22⋊C4—341no—
333C22⋊C4—341no—
334C22⋊C4—341no—
335C22⋊C4—341no—
336C42—171no—
337C42—341no—
338C23—21no—
339C23—21no—
340C23—171no—
341C23—341no—
342C23—341no—
343C23—682no—
344C23—682no—
345C23—1361no—
346C23D4×D1711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D1712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—341no—
357D4—341no—
358D4—341no—
359D4—341no—
360D4—682no—
361D4—681no—
362D4—682no—
363D4—682no—
364D4—1361no—
365D4—1361no—
366D4—1361no—
367D4—1361no—
368D4C2×C22⋊D1712yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—171no—
376C2×C4—171no—
377C2×C4—171no—
378C2×C4—341no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—341no—
382C2×C4—682no—
383C2×C4—1361no—
384C2×C4D4×D1711yescharacteristic
385C2×C4C23×D1711yescharacteristic
386C8—21no—
387C8—341no—
388C8—341no—
389C8C2×C22⋊D1712yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—341no—
400C22—341no—
401C22—341no—
402C22—681no—
403C22—681no—
404C22—681no—
405C22—681no—
406C22—681no—
407C22—681no—
408C22—1364no—
409C22C2×D4×D1711yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—341no—
414C4—341no—
415C4—341no—
416C4—681no—
417C4—681no—
418C4#1058769111yescharacteristic
419C4C22×C22⋊D1711yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—681no—
424C2—1361no—
425C2—1361no—
426C2D4×C22⋊D1711yescharacteristic
427C2#1176360611yescharacteristic
428C2#1176361111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×#135C2
socleC22×C17C22×D4
hypercenterC2×#135D17
hypocenterC17#3268
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#10587600C2semidirect
2. #11763478C2semidirect
3. #11763447C2semidirect
4. #11763195C2semidirect
5. #11763052C2semidirect
6. #11763045C2semidirect
7. #11763033C2semidirect
8. #11763002C2semidirect
9. #11762970C2semidirect
10. #11762954C2semidirect
11. #11762954C2semidirect
12. #11762941C2semidirect
13. #11762910C2semidirect
14. #11762880C2semidirect
15. C2×C17×#135C2semidirect
17. C2×#41×C17C22semidirect
21. C2×Q16×C17C22semidirect
23. C2×SD16×C17C22semidirect
25. C22×C8×C17C22semidirect
28. Q8×Dic17C22semidirect
30. C2×#63230C22semidirect
32. #10587626C22semidirect
42. #10587616C22semidirect
50. #10587506C22semidirect
51. #10587506C22semidirect
53. #10587503C22semidirect
54. C22⋊#3622C22semidirect
56. #10587492C22semidirect
58. #10587489C22semidirect
59. #10587486C22semidirect
60. #10587485C22semidirect
94. SD16×C17D4semidirect
207. C2×#135D17semidirect
232. C2×#41C2×D17semidirect
236. C2×Q16C2×D17semidirect
240. C2×SD16C2×D17semidirect
246. C22×C8C2×D17semidirect
275. C17#3268semidirect
311. SD16C22⋊D17semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#105876002
#117634782
#117634472
#117631952
#117630522
#117630452
#117630332
#117630022
#117629702
#117629542
#117629412
#117629102
#117628802
C2×C17×#1352
#326817